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When Maths Doesnt Work: What we learn from the Prisoners Dilemma Tony Mann, 16 February 2015 Paradoxes and Games 16 February When Maths Doesn't Work: What we learn from the Prisoners' Dilemma

16 March Two Losses Make a Win: How a Physicist Surprised Mathematicians A personal account Theory of Games and Economic Behaviour John von Neumann and Oskar Morgenstern, 1944 Sir Peter Swinnerton-Dyer Lecture, Quintic Society, c.1976

Rock, Paper, Scissors Rock, Paper, Scissors Opponents choice My choice Rock Paper

Scissors Rock Paper Scissors Draw Win Lose

Lose Draw Win Win Lose Draw Penalty Kick

Penalty Kick Goalkeepers choice Strikers choice Left Right

Goes left Goes right Waits 0.95 0.4 0.4

0.95 0.7 0.7 Rock, Paper, Scissors Using Maths Assume the worst Opponent will exploit our strategy if

they can We want to minimise our maximum possible expected loss Rock, Paper, Scissors Opponents choice Rock Paper

Scissors Rock 0 -1 1

Paper 1 0 -1 -1

1 0 My choice Scissors My choice Rock

Paper Scissors Rock 0 1 -1 Opponents choice Paper

Scissors -1 0 1 1 -1 0 I choose Rock with probability , Paper with probability

and Scissors with probability . Assume is bigger than either of the others If my opponent plays Paper every time, my expected outcome is 1x1x which is negative I do best to make , and all equal to 1/3. , http://www.nytimes.com/interactive/ science/rock-paper-scissors.html

Variants French: Pierre, papier, ciseaux, puits Well beats rock and scissors but loses to paper German: Adds Bull which drinks well, eats paper, stabbed by scissors and crushed by rock , Rock paper scissors lizard Spock

Zero-Sum Games Sum of both players outcomes is zero One players winnings exactly match the other players loss The Prisoners Dilemma Sheriff seeking bank

robbers Arrests two cowboys Offers each a deal The Prisoners Dilemma If neither admits crime, both get two years in gaol for another offence If one admits it and implicates the other, he gets off with six

months and other gets 20 years If both admit it, 15 years each Prisoners Dilemma My choice C (Co-operate) D (Defect) Partners choice

C D (Co-operate) (Defect) -2 -20 -0.5

-15 Prisoners Dilemma My choice C (Co-operate) D (Defect) Partners choice

C D (Co-operate) (Defect) -2 -20 -0.5

-15 The Prisoners Dilemma Mathematician-cowboys get 15 years each Non-mathematical cowboys get 2 years each

Examples of the Prisoners Dilemma 1970s wage demands and inflation Whooping-cough vaccination The Tragedy of the Commons The Prisoners Dilemma Maths doesnt work?

It is rational to be selfish? We need a nanny state? The Cold War Cold War Arms Race Us Use resources for other things

Expand nuclear arsenal Other superpower Use resources Expand for other things nuclear arsenal Military

stalemate: social benefits Military disadvantage Military advantage Military

stalemate: less social benefit Cold War Arms Race Us Use resources for other things Expand nuclear arsenal

Other superpower Use resources Expand for other things nuclear arsenal Military stalemate: social benefits

Military disadvantage Military advantage Military stalemate: less social benefit

Tosca Tosca Toscas viewpoint Scarpia Tosca Co-operate (doesnt kill Scarpia)

Defect (kills Scarpia) Co-operate Defect (fake bullets) (real bullets) OK Worst

Best Pretty bad Tosca Scarpias viewpoint Tosca Co-operate (doesnt kill Scarpia)

Defect (kills Scarpia) Co-operate (fake bullets) OK

Worst Defect (real bullets) Best Pretty bad Scarpia

Tosca Scarpias viewpoint Tosca Co-operate (doesnt kill Scarpia) Defect (kills

Scarpia) Co-operate (fake bullets) OK Worst Defect

(real bullets) Best Pretty bad Scarpia Prisoners Dilemmas

Nobel Prizes Twelve game theorists have won a Nobel Prize Steven J. Brams Jane Austen and Game Theory

Golden Balls Golden Balls Opponents Choice Split Steal Split

Half of Jackpot 0 Steal Whole of Jackpot 0

Golden Balls Opponents Choice Split Steal Split Half of Jackpot

0 Steal Whole of Jackpot 0 Biology

Birds Hawks and Doves Food worth 50 units Birds may fight loser suffers injury worth 100 units Competitive display cost 10 units to each bird Hawk always fights, dove

concedes food to a hawk Hawks and Doves Hawk Dove Hawk x 50 x 100

= 25 + 50 Dove 0 ( x 50 + x 0) 10 = +15

Hawks and Doves Evolutionary stable strategy: 58% are hawks No incomers with a different strategy can exploit the colony Robert Axelrod

Scientific American, May 1983 Iterated Prisoners Dilemma Computer tournaments Invited people to submit strategies Winning Strategy Anatol Rapoports Tit for Tat Start by Co-operating

Then do whatever opponent did on previous round Tit for Tat Never outscores its immediate opponent! But stimulates mutually rewarding behaviour from opponents Tit for Tat

Nice never first to defect Forgiving doesnt bear grudges Retaliatory if you defect it defects back Clear easy for opponent to understand Game Theory Rather than the Tragedy of the Commons defining the human condition, the Prisoners Dilemma

explains the evolution of co-operation Matt Ridley Matt Ridley The Tragedy of the Commons Matt Ridley

Anger The Science of Co-operation Our hope for international co-operation? A Personal Story of the Prisoners Dilemma A refutation of game theory? A fascinating paradox?

Solution to the problem of the evolution of altruism? Tool for exploring co-operation? Our best hope for the future? The Prisoners Dilemma Theres still so much more left to find out. We have only explored a small subset of this extraordinary

game Our analysis of how to solve the Dilemma will never be completed. This Dilemma has no end. Martin Nowak Thank you for listening [email protected] @Tony_Mann

Acknowledgments and picture credits Thanks to Noel-Ann Bradshaw and everyone at Gresham College Slide design Aoife Hunt Picture credits Images from Wikimedia Commons they are used under a Creative Commons licence: full details can be found at Wikimedia Commons Photograph of lecturer: Noel-Ann Bradshaw; T-shirt: www.thinkgeek.com Edinburgh: Tilmandralle, Wikimedia Commons, public domain Peterhouse, Cambridge: Robert Edwards, Wikimedia Commons

University of Greenwich: University of Greenwich John von Neumann: Los Angeles National Laboratory., Wikimedia Commons. http://www.lanl.gov/resources/web-policies/copyright-legal.php Sir Peter Swinnerton-Dyer: Renate Schmid, Wikimedia Commons Rock, paper, scissors: Photo by Fluff, modified by Sertion, Wikimedia Commons Penalty kick by Ryan Valentine (Wrexham), 2007: Markbarnes, Wikimedia Commons Rock, paper, scissors, lizard, spock: DMacks, Wikimedia Commons Cowboy, sheriff, sheep, cows: Microsoft clip art Greenhouse George mushroom cloud: National Nuclear Security Administration, public domain Dnepr rocket: ISC Kosmotras, Wikimedia Commons Tosca: EJ Wheeler cartoon, Punch 1888, public domain

Tosca and Scarpia pre-1914 production, Metropolitan Opera House, New York, Wikipedia, public domain Jasper Carrott: HugoVK, Wikimedia Commons John Maynard Smith: Web of Stones, Wikimedia Commons Nobel Prize: Wikimedia Commons, public domain Steven J. Brams: Eva Brams, Wikimedia Commons Matt Ridley: Tara Hunt, Wikimedia Commons Northern Rock: Alex Gunningham, Wikimedia Commons Anger: fresco from Saint Nicolas church in Cukovets, Bulgaria: Edal Anton Lefterov, Wikimedia Commons Martin Nowak: Filip Antoni Malinowski, Wikimedia Commons Mauna Loa Carbon Dioxide graph: Smhur / Wikimedia Commons

Further Reading John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behaviour (Princeton, 60th anniversary edition, 2004) Steven J. Brams, Game Theory and the Humanities: Bridging Two Worlds (MIT Press, 2011) Robert Axelrod, The Evolution of Co-operation (Penguin 1990; Basic 2009) Matt Ridley, The Origins of Virtue (Penguin, 1997) Martin Nowak with Roger Highfield: Super Cooperators: Evolution,

Altruism and Human Behaviour, or Why We Need Each Other to Succeed (Canongate, 2011)

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